Quadratic ideals, indefinite quadratic forms and some specific diophantine equations

• Main Authors: Ahmet Tekcan, Seyma Kutlu
• Format: Article
• Language: English
• Published: Sociedade Brasileira de Matemática 2018-07-01
• Series: Boletim da Sociedade Paranaense de Matemática
• Subjects: Quadratic ideals; indefinite quadratic forms; Diophantine equations
• Online Access: http://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/30799

Let $k\geq 1$ be an integer and let $P=k+2,Q=k$ and $D=k^{2}+4$. In this paper, we derived some algebraic properties of quadratic ideals $I_{\gamma}$ and indefinite quadratic forms $F_{\gamma }$ for quadratic irrationals $\gamma$, and then we determine the set of all integer solutions of the Diophantine equation $F_{\gamma }^{\pm k}(x,y)=\pm Q$.